C₅×#10391 #11933111

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#10391

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#393
commutator, Frattini, cosocle	C₂⁵	C₂³×C₅
prime radical	C₂⁵×C₅	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁵×C₅ ◁ G

542 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7)(8,12)(9,20,22,13)(10,14)(11,16)(15,24)(17,32,21,29)(18,27)(19,28)(23,30,36,31)(25,33,37,35)(26,34);(6,8)(7,12)(9,21,22,17)(10,19)(11,18)(13,29,20,32)(14,28)(15,26)(16,27)(23,35,36,33)(24,34)(25,30,37,31);(6,9,11,23)(7,13,16,31)(8,17,18,33)(10,22,24,36)(12,25,27,29)(14,20,15,30)(19,21,34,35)(26,32,28,37)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3379);